This section provides background information related to the present disclosure which is not necessarily prior art. This section also provides a general summary of the disclosure, and is not a comprehensive disclosure of its full scope or all of its features.
Unlike robust control, which chooses control gains based on a prior, fixed level of modeling uncertainty, adaptive control algorithms tune the feedback gains in response to the true dynamical system (or “plant”), and commands and disturbances (collectively “exogenous signals”). Generally speaking, adaptive controllers require less prior modeling information than robust controllers, and thus can be viewed as highly parameter-robust control laws. The price paid for the ability of adaptive control laws to operate with limited prior modeling information is the complexity of analyzing and quantifying the stability and performance of the closed-loop system, especially in light of the fact that adaptive control laws, even for linear plants, are nonlinear.
Stability and performance analysis of adaptive control laws often entails assumptions on the dynamics of the plant. For example, a widely invoked assumption in adaptive control is passivity, which is restrictive and difficult to verify in practice. A related assumption is that the plant is minimum phase, which may entail the same difficulties. In fact, sampling may give rise to nonminimum-phase zeros whether or not the continuous-time system is minimum phase, which must ultimately be accounted for by any adaptive control algorithm implemented digitally in a sampled-data control system. Beyond these assumptions, adaptive control laws are known to be sensitive to unmodeled dynamics and sensor noise, which necessitates robust adaptive control laws.
In addition to these basic issues, adaptive control laws may entail unacceptable transients during adaptation, which may be exacerbated by actuator limitations. In fact, adaptive control under extremely limited modeling information, such as uncertainty in the sign of the high-frequency gain, may yield a transient response that exceeds the practical limits of the plant. Therefore, the type and quality of the available modeling information as well as the speed of adaptation must be considered in the analysis and implementation of adaptive control laws.
Adaptive control laws have been developed in both continuous-time and discrete-time settings. In the present application we consider discrete-time adaptive control laws since these control laws can be implemented directly in embedded code for sampled-data control systems without requiring an intermediate discretization step that may entail loss of stability margins.
According to some prior art, references on discrete-time adaptive control include a discrete-time adaptive control law with guaranteed stability developed under a minimum-phase assumption. Extensions based on internal model control and Lyapunov analysis also invoke this assumption. To circumvent the minimum-phase assumption, the zero annihilation periodic control law uses lifting to move all of the plant zeros to the origin. The drawback of lifting, however, is the need for open-loop operation during alternating data windows. An alternative approach, is to exploit knowledge of the nonminimum-phase zeros. Knowledge of the nonminimum-phase zeros is used to allow matching of a desired closed-loop transfer function, recognizing that minimum-phase zeros can be canceled but not moved, whereas nonminimum-phase zeros can neither be canceled nor moved. Knowledge of a diagonal matrix that contains the nonminimum-phase zeros is used within a MIMO direct adaptive control algorithm. Finally, knowledge of the unstable zeros of a rapidly sampled continuous-time SISO system with a real nonminimum-phase zero is used in some instances.
Motivated by the adaptive control laws given in some instances, the goal of the present application is to develop a discrete-time adaptive control law that is effective for nonminimum-phase systems. In particular, we present an adaptive control algorithm that extends the retrospective cost optimization approach. This extension is based on a retrospective cost that includes control weighting as well as a learning rate, which can be used to adjust the rate of controller convergence and thus the transient behavior of the closed-loop system. Unlike some instances, which use a gradient update, the present application uses a Newton-like update for the controller gains as the closed-form solution to a quadratic optimization problem. No off-line calculations are needed to implement the algorithm or control system. A key aspect of this extension is the fact that the required modeling information is the relative degree, the first nonzero Markov parameter, and nonminimum-phase zeros, if any. Except when the plant has nonminimum-phase zeros whose absolute value is less than the plant's spectral radius, we show that the required zero information can be approximated by a sufficient number of Markov parameters from the control inputs to the performance variables. No matching conditions are required on either the plant uncertainty or disturbances.
In some embodiments, a goal of the present application is to develop the RCF adaptive control algorithm and demonstrate its effectiveness for handling nonminimum-phase zeros. To this end we consider a sequence of examples of increasing complexity, ranging from SISO, minimum-phase plants to MIMO, nonminimum-phase plants, including stable and unstable cases. We then revisit these plants under off-nominal conditions, that is, with uncertainty in the required plant modeling data, unknown latency, sensor noise, and saturation. These numerical examples provide guidance into choosing the design parameters of the adaptive control law in terms of the learning rate, data window size, controller order, modeling data, and control weightings.
According to the principles of the present teachings, a discrete-time adaptive control law or algorithm for stabilization, command following, and disturbance rejection that is effective for systems that are unstable, MIMO, and/or nonminimum phase. The adaptive control algorithm includes guidelines concerning the modeling information needed for implementation. This information includes the relative degree, the first nonzero Markov parameter, and the nonminimum-phase zeros. Except when the plant has nonminimum-phase zeros whose absolute value is less than the plant's spectral radius, the required zero information can be approximated by a sufficient number of Markov parameters. No additional information about the poles or zeros need be known. We present numerical examples to illustrate the algorithm's effectiveness in handling systems with errors in the required modeling data, unknown latency, sensor noise, and saturation.
Further areas of applicability will become apparent from the description provided herein. The description and specific examples in this summary are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure.
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